\documentclass{article}
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\DeclarePairedDelimiter\norm{\lVert}{\rVert}
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\def\P{\mathcal{P}}
\def\E{\mathbb{E}}
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\newtheorem{lemma}{Lemma}
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\title{Not feasible}
\author{Feng Zhao}
\begin{document}
\maketitle
If we consider $\E[\min_{\hat{w}}\norm{Y - X \bar{w} - \epsilon \xi (X \bar{w} + \epsilon X \hat{w})}^2]$ where
$\bar{w} = X^T Y$. We can simplify this expression to
$\E[\norm{Y-XX^TY}^2] + \epsilon^2 \E[\min_{\hat{w}}\{\norm{\xi(XX^TY)}^2 - 2(\nabla \xi(XX^TY) X\hat{w})^T(Y-XX^TY)\}] + \min_{\hat{w}}C(\hat{w})\epsilon^3 + o(\epsilon^3)$. 

We choose $(\nabla \xi(XX^TY) X)^T (Y-XX^TY) = 0$, 
thus the coefficient of $\epsilon^2$ is zero.
Therefore, we need to optimize $C(\hat{w}) = \xi(XX^TY)^T \nabla \xi(XX^TY) X\hat{w} - 2(Y-XX^TY)^T\hat{w}X^T\nabla^2\xi(XX^TY)X\hat{w}$, which is quadratic with $\hat{w}$.
\end{document}